interest rates - Interpolating probabilities of default

I have a table of cumulative probabilities of default of industrial bonds, in time and credit rating. It is similar to S&P whitepaper here. Basically, it looks like this (sample numbers):

Years | AAA   | AA    | A    | ... | C
  1   | 0.01% | 0.04% | 0.09 | ...
  ...

  30  | 1%    | 5%    | 8%   | ...

This data has gaps both in time and in credit rating. Is there any standard methodology on how to do such interpolations/extrapolations or perhaps a paper/book I can read on the subject?

On a related note, what if the numbers in the table are interest rates for the corresponding bonds - is there a methodology for that?

Thank you very much.

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Answer

I believe that your problem can be formulated as:

Find PD matrix that is as close as possible to a given PD matrix (result of some previous calibration, or the matrix computed using average hazard rate, or any other "target", or the penalty on non-smoothness) subject to the following constraints:

1. The values that are given must be matched exactly


2. Monotonicity constraints (in both time and rating) must be satisfied, and so are 0 and 1 bounds.

This fits the definition of quadratic programming. Matlab has got it [implemented].2

Quadratic programming is a method that allows you to find the best possible answer. However, if there are not too many gaps, and being close to the "target" is not essential, you can interpolate/extrapolate them manually, subject to constraints